Abstract: The invention relates to a finish cooling unit for a container glass machine for molding a glass container, comprising at least one plunger cylinder having a plunger cylinder cover (6) and a parison mold (4). Said plunger cylinder cover (6) has at least one feed line and at least one channel (PK) with an outlet through which a cooling medium (KM) is guided and exits the channel (PK) in the plunger cylinder cover (6). The cooling medium (KM) for the finish is supplied from a source separate from the source for cooling the parison mold in order to cool the finish of the glass container to be molded independent of the cooling of the parison mold (4). The invention also relates to a method for cooling the finish of a glass container during parison molding in a container glass machine, whereby the finish of the glass container is cooled using a control or adjustment of the pressure of the cooling medium (KM) independent of the cooling of the parison mold.